118 research outputs found
Spectral asymptotics for the Schr\"odinger operator on the line with spreading and oscillating potentials
This study is devoted to the asymptotic spectral analysis of multiscale
Schr\"odinger operators with oscillating and decaying electric potentials.
Different regimes, related to scaling considerations, are distinguished. By
means of a normal form filtrating the oscillations, a reduction to a
non-oscillating effective Hamiltonian is performed
A new class of two-layer Green-Naghdi systems with improved frequency dispersion
We introduce a new class of Green-Naghdi type models for the propagation of
internal waves between two (1+1)-dimensional layers of homogeneous, immiscible,
ideal, incompressible, irrotational fluids, vertically delimited by a flat
bottom and a rigid lid. These models are tailored to improve the frequency
dispersion of the original bi-layer Green-Naghdi model, and in particular to
manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its
precision in the sense of consistency. Our models preserve the Hamiltonian
structure, symmetry groups and conserved quantities of the original model. We
provide a rigorous justification of a class of our models thanks to
consistency, well-posedness and stability results. These results apply in
particular to the original Green-Naghdi model as well as to the Saint-Venant
(hydrostatic shallow-water) system with surface tension.Comment: to appear in Stud. Appl. Mat
The multilayer shallow water system in the limit of small density contrast
Minor modifications with respect to v1.International audienceWe study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial data, accounting for rapidly propagating "acoustic" waves (here, the so-called barotropic mode) which interact only weakly with the "incompressible" component (here, baroclinic)
A note on the well-posedness of the one-dimensional multilayer shallow water model
In this short note, we prove by elementary means that the one-dimensional multilayer shallow-water (or Saint-Venant) model for density-stratified fluids is well-posed, provided that (i) the density stratification is stable (i.e. the denser fluid is deeper); (ii) each layer has non-vanishing depth; (iii) the shear velocities are small enough
Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes
Let denote a periodic function on the real line. The Schr\"odinger
operator, , has spectrum equal to
the union of closed real intervals separated by open spectral gaps. In this
article we study the bifurcation of discrete eigenvalues (point spectrum) into
the spectral gaps for the operator , where is
spatially localized and highly oscillatory in the sense that its Fourier
transform, is concentrated at high frequencies. Our
assumptions imply that may be pointwise large but is
small in an average sense. For the special case where
with smooth, real-valued, localized in
, and periodic or almost periodic in , the bifurcating eigenvalues are at
a distance of order from the lower edge of the spectral gap. We
obtain the leading order asymptotics of the bifurcating eigenvalues and
eigenfunctions. Underlying this bifurcation is an effective Hamiltonian
associated with the lower edge of the spectral band:
where is the Dirac distribution,
and effective-medium parameters are
explicit and independent of . The potentials we consider are a
natural model for wave propagation in a medium with localized, high-contrast
and rapid fluctuations in material parameters about a background periodic
medium.Comment: To appear in SIAM Journal on Mathematical Analysi
Asymptotic shallow water models for internal waves in a two-fluid system with a free surface
In this paper, we derive asymptotic models for the propagation of two and
three-dimensional gravity waves at the free surface and the interface between
two layers of immiscible fluids of different densities, over an uneven bottom.
We assume the thickness of the upper and lower fluids to be of comparable size,
and small compared to the characteristic wavelength of the system (shallow
water regimes). Following a method introduced by Bona, Lannes and Saut based on
the expansion of the involved Dirichlet-to-Neumann operators, we are able to
give a rigorous justification of classical models for weakly and strongly
nonlinear waves, as well as interesting new ones. In particular, we derive
linearly well-posed systems in the so called Boussinesq/Boussinesq regime.
Furthermore, we establish the consistency of the full Euler system with these
models, and deduce the convergence of the solutions.Comment: 32 pages, 4 figure
Homogenized description of defect modes in periodic structures with localized defects
A spatially localized initial condition for an energy-conserving wave
equation with periodic coefficients disperses (spatially spreads) and decays in
amplitude as time advances. This dispersion is associated with the continuous
spectrum of the underlying differential operator and the absence of discrete
eigenvalues. The introduction of spatially localized perturbations in a
periodic medium leads to defect modes, states in which energy remains trapped
and spatially localized. In this paper we study weak, localized perturbations
of one-dimensional periodic Schr\"odinger operators. Such perturbations give
rise to such defect modes, and are associated with the emergence of discrete
eigenvalues from the continuous spectrum. Since these isolated eigenvalues are
located near a spectral band edge, there is strong scale-separation between the
medium period and the localization length of the defect mode. Bound states
therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave
envelope", which is governed by a homogenized Schr\"odinger operator with
associated effective mass, depending on the spectral band edge which is the
site of the bifurcation. Our analysis is based on a reformulation of the
eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch
transform and a Lyapunov-Schmidt reduction to a closed equation for the
near-band-edge frequency components of the bound state. A rescaling of the
latter equation yields the homogenized effective equation for the wave
envelope, and approximations to bifurcating eigenvalues and eigenfunctions.Comment: The title differs from version 1. To appear in Communications in
Mathematical Science
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